1. Field of the Invention
The present invention generally relates to method, system and software product used in finite element analysis, more particularly to numerical simulation of the structural behaviors of highly compressible material such as foam under loading conditions.
2. Description of the Related Art
Finite element analysis (FEA) is a computerized method widely used in industry to model and solve engineering problems relating to complex systems. FEA derives its name from the manner in which the geometry of the object under consideration is specified. With the advent of the modern digital computer, FEA has been implemented as FEA software. Basically, the FEA software is provided with a model of the geometric description and the associated material properties at each point within the model. In this model, the geometry of the system under analysis is represented by solids, shells and beams of various sizes, which are called elements. The vertices of the elements are referred to as nodes. The model is comprised of a finite number of elements, which are assigned a material name to associate with material properties. The model thus represents the physical space occupied by the object under analysis along with its immediate surroundings. The FEA software then refers to a table in which the properties (e.g., stress-strain constitutive equation, Young's modulus, Poisson's ratio, thermo-conductivity) of each material type are tabulated. Additionally, the conditions at the boundary of the object (i.e., loadings, physical constraints, etc.) are specified. In this fashion a model of the object and its environment is created.
FEA is becoming increasingly popular with automobile manufacturers for optimizing both the aerodynamic performance and structural integrity of vehicles. Similarly, aircraft manufacturers rely upon FEA to predict airplane performance long before the first prototype is built. Rational design of semiconductor electronic devices is possible with Finite Element Analysis of the electrodynamics, diffusion, and thermodynamics involved in this situation. FEA is utilized to characterize ocean currents and distribution of contaminants. FEA is being applied increasingly to analysis of the production and performance of such consumer goods as ovens, blenders, lighting facilities and many plastic products. In fact, FEA has been employed in as many diverse fields as can be brought to mind, including plastics mold design, modeling of nuclear reactors, analysis of the spot welding process, microwave antenna design, simulating of car crash and biomedical applications such as the design of prosthetic limbs. In short, FEA is utilized to expedite design, maximize productivity and efficiency, and optimize product performance in virtually every stratum of light and heavy industry. This often occurs long before the first prototype is ever developed.
The finite element analysis method is described in detail by Thomas J. R. Hughes in “The Finite Element Method” (1987), published by Prentice-Hall, Inc., New Jersey, which is incorporated herein by reference in its entirety. Generally, FEA begins by generating a finite element model of a system. In this model, a subject structure is reduced into a number of node points which are connected together to form finite elements. Governing equations of motion are written in a discrete form, where the motions of each node point are the unknown part of the solution. A simulated load or other influence is applied to the system and the resulting effect is analyzed using well known mathematical methods.
FEA software can be classified into two general types, implicit FEA software and explicit FEA software. Implicit FEA software uses an implicit equation solver to solve a system of coupled linear equations. Such software is generally used to simulate static or quasi-static problems. Explicit FEA software does not solve coupled equations but explicitly solves for each unknown assuming them uncoupled. Explicit FEA software usually uses central difference time integration which requires very small solution cycles or time steps for the method to be stable and accurate. Explicit FEA software is generally used to simulate short duration events where dynamics are important such as impact type events.
One of the most challenging FEA tasks is to simulate an impact event such as car crash. The highly non-linear behavior of the structural materials must be numerically simulated accurately, realistically and efficiently. A number of materials such as steel, aluminum, foam and rubber that used in an automobile must be included in such FEA software. Many components used in an automobile are made of materials such as foam and rubber, the simulation of the structural responses of these materials becomes very important for the overall accuracy of an analysis.
Traditionally the structural responses of highly compressible material such as foam have been numerically simulated using the Ogden strain energy function W. A brief summary of the Ogden strain energy function and corresponding engineering/nominal stress σ0 and Cauthy/true stress σ functions are listed in FIG. 1. More details of the Ogden energy function is described by R. W. Ogden in Chapter 7 of the book titled: “Non-linear Elastic Deformations” (1984), published by Ellis Horwood Limited, United Kingdom, which is incorporated herein by reference in its entirety. A number of commercially available FEA software includes these approaches to simulate foam-like material. For example, LS-DYNA, a general purpose three-dimensional non-linear large-deformation FEA software from Livermore Software Technology Corporation, is capable of simulating foam-like material using the Ogden energy function.
Today, there are a number of practical problems associated with the simulation of foam-like material in FEA. To implement the Ogden energy function properly in the FEA software requires engineers to spend a tremendous amount of time to prepare experimental data and then convert them into a set of coefficients to fit a polynomial Ogden function for FEA software. Due to highly non-linear characteristics of this polynomial function, the inexactly fitted function has often resulted. This leads to a lengthy iterative trial-and-error process of modifying the input coefficients to match the behavior of foam-like material in the real world. It is therefore desirable to have a new method to numerically simulate foam-like material more efficiently and effectively.